On the Cohomology of the Classifying Spaces of Projective Unitary Groups

TL;DR

This paper investigates the cohomology ring structure of classifying spaces of projective unitary groups using spectral sequences, revealing new algebraic features and primitive elements up to degree 10.

Contribution

It determines the cohomology ring of $ ext{B}PU_n$ up to degree 10 and identifies special elements related to prime divisors of $n$, advancing understanding of their algebraic topology.

Findings

Cohomology ring structure of $ ext{B}PU_n$ up to degree 10 determined.

Identifies distinguished elements in cohomology related to prime divisors of $n$.

Analyzes primitive elements as a comodule over $H^*(K( ext{Z},2))$.

Abstract

Let $\mathbf{B}PU_{n}$ be the classifying space of $PU_n$, the projective unitary group of order $n$, for $n>1$. We use the Serre spectral sequence associated to a fiber sequence $\mathbf{B}U_n\rightarrow\mathbf{B}PU_n\rightarrow K(\mathbb{Z},3)$ to determine the ring structure of $H^{*}(\mathbf{B}PU_{n}; \mathbb{Z})$ up to degree $10$, as well as a family of distinguished elements of $H^{2p+2}(\mathbf{B}PU_{n}; \mathbb{Z})$, for each prime divisor $p$ of $n$. We also study the primitive elements of $H^*(\mathbf{B}U_n;\mathbb{Z})$ as a comodule over $H^*(K(\mathbb{Z},2);\mathbb{Z})$, where the comodule structure is given by an action of $K(\mathbb{Z},2)\simeq\mathbf{B}S^1$ on $BU_n$ corresponding to the action of taking the tensor product of a complex line bundle and an $n$ dimensional complex vector bundle.